Finite Element Modeling of the Ferroelectroelastic Material Behavior in Consideration of Domain Wall Motions
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Finite element modeling of the ferroelectroelastic material behavior in consideration of domain wall motions Albrecht C. Liskowsky1, Artem S. Semenov1, Herbert Balke1, Robert M. McMeeking2 1 Institute for Solid Mechanics (IFKM) Technical University Dresden, 01069 Dresden, Germany 2 Department of Mechanical and Environmental Engineering and Materials Department University of California, Santa Barbara, CA 93106, U.S.A. ABSTRACT A simulation of the nonlinear electromechanical macroscopic behavior of ferroelectric materials by means of the finite element method is presented. A material point is depicted by a representative volume element, for which homogeneous boundary conditions are valid. The evolution of integral averages over the representative volume element is to homogenize the results. For this homogenization we favor a finite element model in which each Gauss point represents exactly one single crystal. Their number of internal variables is limited to the lattice orientation and the volume fractions of the domains. The former are randomly distributed in space. It is possible to calculate the material behavior for arbitrary coupled and nonlinear electromechanical loading cases, but the model is not effective for the solution of boundary value problems for entire bodies. INTRODUCTION Ferroelectrics are piezoelectric ceramics whose electric polarization in the unit cells can change between several directions, whereby these materials exhibit nonlinear macroscopic properties. In the work under consideration we model by means of finite elements the material behavior of ferroelectrics with tetragonal crystal structure, whose unit cells show six possible polarization directions each. Regions of differing polarizations within the single-crystal grains are designated as domains. The switching, which is the change of the polarization directions of the unit cells, is a dissipative process, by which the walls between the domains shift gradually. By particular polarization ferroelectric materials receive piezoelectric properties for electromechanical loads of lower amplitude that are at any time changeable and precisely adjustable. This happens by the application of high electric fields, and by the associated changes of the volume fractions of the domains within the single crystals. A goal of the work under consideration is the numeric computation of the nonlinear macroscopic behavior of ferroelectric materials. This is all done based on a computation model for the processes at a mesoscopic level. In consideration of Huber et al [2], the progressive motion of the domain walls observed in experiments due to the 90° and 180° switching of the unit cells is taken into account. The interactions between randomly oriented single crystals are computed by means of the finite element method, whereby the single crystals are depicted as Gauss points of three-dimensional finite elements. The knowledge of this nonlinear material behavior is of major importance for the computer-aided construction of ferroelectric sensors and actuator
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